On Frobenius semisimplicity in Hida families

Let $p\ge 5$ be a prime and $\mathrm{\ell }\mathrm{\ne }p$ be a prime not dividing the tame level of a $p$-ordinary Hida family. We prove that the actions of the Frobenius element at $\mathrm{\ell }$ on the Galois representations attached to almost all arithmetic specializations are semisimple and non-scalar. If $k_f$ denotes the weight of a cusp form $f(z)={\sum }_{n\ge 1}{a}_{\mathrm{\ell }}(f)e^{2\pi inz}$, then the inequality $$\mathrm{\mid }{a}_{\mathrm{\ell }}(f)\mathrm{\mid }\le 2{\mathrm{\ell }}^{(k_f-1)\mathrm{/}2},$$ predicted by the Ramanujan conjecture, is a strict inequality for almost all members $f$ of the family.